A theorem of A. G. Vitushkin

An important condition for the coincidence of the algebras R(E) and C(E) on a compactum was found by A. G. Vitushkin. In this note we give a simple proof of Vitushkin's theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 178–181, 1984.     

A formal Frobenius theorem and argument shift

A formal Frobenius theorem, which is an analog of the classical integrability theorem for smooth distributions, is proved and applied to generalize the argument shift method of A. S. Mishchenko and A. T. Fomenko to finite-dimensional Lie algebras over any field of characteristic zero. A completeness criterion for a commutative set of polynomials constructed by the formal argument shift method is obtained.     

Sturm's theorem for equations with delayed argument

Sturm's type theorems on separation of zeros of solutions are proved for second order linear differential equations with delayed argument.     

On the exactness of a theorem of G.A. Fomin

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On a theorem of E. Helly

E. Helly's theorem asserts that any bounded sequence of monotone real functions contains a pointwise convergent subsequence. We reprove this theorem in a generalized version in terms of monotone functions on linearly ordered sets. We show that the cardinal number responsible for this generalization is exactly the splitting number. We also show that a positive answer to a problem of S. Saks is obtained under the assumption of the splitting number being strictly greater than the first uncountable cardinal.

     

On a theorem of E. Cartan

Summary The object of this note is to prove: — Let G be a connected, locally compact subgroup of an analytic group H modelled on a Banach space. Then G itself is a finite dimensional analytic subgroup of H.     

A limit theorem for the argument of zeta-functions of certain cusp forms

We obtain a limit theorem for the modulus of the argument of zeta-functions near the critical line of normalized eigenforms. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 501–512, October–December, 2006.     

On a theorem of G. Szegö

It is shown that the precise result of G. Szegö on the asymptotic behavior of the Toeplitz determinants D_n(f), generated by the nonnegative summable function off() holds if Inf L₁ (–, ).Translated from Matematicheskie Zametki, Vol. 3, No. 6, pp. 693–702, June, 1968.     

The Erdös-Wintner theorem for additive functions of a rational argument

Vilnius University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 30, No. 2, pp. 405–415, April–June, 1990.     

On a combinatorial theorem related to a theorem of G. Szegö

In 1954, M. Kac discovered a probabilistic interpretation of a theorem of G. Szegö of Toeplitz matrices and demonstrated that this theorem can be provedin an elementary way by using a combinatorial identify of G. A. Hunt. In this paper Hunt's combinatorial identity is derived from a more general combinatorial result.     

An extension of a theorem of G. Tallini

A Tallini set in a projective space P is a set Q of points of P such that each line not contained in Q intersects Q in at most two points. We prove that if P is a finite projective space with odd order q > 3 and dimension d > 2 and if |Q| > q^{d ? 1} + 2q^{d ? 3} + q^{d ? 4} + … + 1, then Q is essentially an orthogonal quadric. The proof of this theorem is based on a characterization of the orthogonal quadrics in every finite dimensional projective space (with possibly infinite order).     

A proof of the butterfly theorem by an argument from statics

The butterfly theorem is proved by assigning point masses to the four vertices of the wings and using the distributive property of the mass centre of a mechanical system.